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This paper considers traces at the initial time for solutions of evolutionequations with local or non-local derivatives in vector-valued $L_p$ spaceswith $A_p$ weight. To achieve this, we begin by introducing a generalized realinterpolation method. Within the framework of generalized interpolation theory,we make use of stochastic process theory and two-weight Hardy's inequality toderive our trace and extension theorems. Our results encompass findingsapplicable to time-fractional equations with broad temporal weight functions.

On the trace theorem to Volterra-type equations with local or non-local derivatives